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What is a confirmation?

When you make a bitcoin transaction, it goes in to a "holding area" on the bitcoin network. Whilst a transaction is in this holding area (called the transaction pool), it's possible to try and send the same bitcoins somewhere else.

Therefore, each miner on the bitcoin network tries to add all of the transactions in the transaction pool on to a big file of "confirmed" trasactions (called the blockchain). It takes a lot of processing power to achieve this, and when is has been done, it is very difficult to undo.

Every time a new batch of transactions gets added to the blockchain (on top of the transacion you made), the number of confirmations for your transaction increases.

So when a transaction gets confirmed, it cannot be undone easily.

The more confirmations a transaction has, the more impossible it is to reverse. 1 confirmation is usually enough, but you can wait for 2 or more to be sure.

If you want to see how many confirmations your transaction has, check the bitcoin address (the one you're sending bitcoins to) at blockchain.info.

How can a casino accept a deposit with 0 confirmations?

Because they wait for the confirmation before allowing you to withdraw.

If a casino accepts instant bitcoin deposits, then they will credit your account as soon as they see the transaction appear on the network, and you can place bets immediately.

So if you make a transaction, don't be horrified if it takes 30 minutes or so for that first confirmation -- it's just the way it goes occassionally.

The difficulty of finding a correct number is set by the difficulty It changes every 2016 blocks to try and create an average of 10 minutes between adding new blocks to the blockchain (i.e. another transaction confirmation).